Part III

Greek Mathematics From 500 BCE to 500 CE

During the millennium from 500 BCE to 500 CE, mathematics, especially geometry, was imported into Greece, became mixed with the speculations of the Greek philosophers and developed into a body of knowledge that was unique in its time. The center of gravity gradually shifted from the commercial Ionian colonies in the early period (along with their colonies in Italy and Sicily) to Athens in the fifth century, reaching its peak in the third century at Alexandria, Egypt. Along with this new, formal geometry, some of the earlier, more practically oriented geometry survived and revived after the time of the three greatest geometers of antiquity (Euclid, Archimedes, and Apollonius). This practical geometry was applied to produce one great astronomical treatise, the Almagest of Claudius Ptolemy, which became a standard reference for the next thousand years. After Ptolemy, the potential of the Euclidean methods was nearly exhausted, and there is little that is original in the last few centuries we are going to discuss. The one exception comes from number theory, rather than geometry, and it is the invention of symbolic algebraic notation by Diophantus in order to solve problems involving the arithmetic properties of figurate numbers. The next 11 chapters give a sketch of some of the highlights of this long period of development.

Contents of Part III

1. Chapter 8 (An Overview of Ancient Greek Mathematics) gives a survey of the whole period and summarizes the sources on which our knowledge of it is based.

2. Chapter 9 (Greek Number Theory) looks at ancient Greek number theory through the works of Euclid, Nicomachus, and Diophantus.

3. Chapter 10 (Fifth-Century Greek Geometry) presents a hypothetical scenario for the development of geometry up to the mid-fifth century BCE.

4. Chapter 11 (Athenian Mathematics I: The Classical Problems) brings the development of geometry in Athens to the end of the fourth century BCE.

5. Chapter 12 (Athenian Mathematics II: Plato and Aristotle) discusses the connection of this geometry with the philosophies of Plato and Aristotle.

6. Chapter 13 (Euclid of Alexandria) analyzes the Elements and looks briefly at some other works by Euclid.

7. Chapter 14 (Archimedes of Syracuse) is devoted to the works of Archimedes.

8. Chapter 15 (Apollonius of Perga) discusses the extensive treatise on conic sections by Apollonius of Perga.

9. Chapter 16 (Hellenistic and Roman Geometry) discusses the isoperimetric problems studied by Zenodorus and the return of metric concepts to geometry in the work of Heron of Alexandria.

10. Chapter 17 (Ptolemy's Geography and Astronomy) is devoted to the geographical and astronomical treatises of Claudius Ptolemy.

11. Chapter 18 (Pappus and the Later Commentators) summarizes the work of the later commentators Pappus, Theon of Alexandria, and Theon's daughter Hypatia.